We consider the system AotoYamada_05__024.

  Alphabet:

    cons : [a * b] --> b 
    map : [a -> a * b] --> b 
    nil : [] --> b 

  Rules:

    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, cons, map}, and the following precedence: map > cons > @_{o -> o}

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  map(F, _|_) >= _|_ 
  map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y)) 

With these choices, we have:

  1] map(F, _|_) >= _|_  by (Bot) 

  2] map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y))  because [3], by definition 
  3] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [4] and [11], by (Copy) 
  4] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [5] and [7], by (Copy) 
  5] map*(F, cons(X, Y)) >= F  because [6], by (Select) 
  6] F >= F  by (Meta) 
  7] map*(F, cons(X, Y)) >= X  because [8], by (Select) 
  8] cons(X, Y) >= X  because [9], by (Star) 
  9] cons*(X, Y) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [12] and [13], by (Stat) 
  12] F >= F  by (Meta) 
  13] cons(X, Y) > Y  because [14], by definition 
  14] cons*(X, Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

We can thus remove the following rules:

  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  map(F, nil) >? nil 

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[nil]] = _|_ 

We choose Lex = {} and Mul = {map}, and the following precedence: map

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  map(F, _|_) > _|_ 

With these choices, we have:

  1] map(F, _|_) > _|_  because [2], by definition 
  2] map*(F, _|_) >= _|_  by (Bot) 

We can thus remove the following rules:

  map(F, nil) => nil 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.